% Paper: The fundamental lemma for $Sp(4)$
% Author: Thomas C. Hales
% Date: July 10, 1995
% Format: Ams-Tex, amsppt
% published by the Proc. of the AMS, Vol 125, No 1, Jan 1997, 301--308
% Copyright 1997, AMS.
% Reference Number:  PROC3546.
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\begin{document}

\title{The Fundamental Lemma for $Sp(4)$}
\author{Thomas C. Hales}
\address{Ann Arbor, Michigan}
\thanks{Research supported by the National Science Foundation}
\date{Recieved by the editors February 14, 1995 and, in revised form,
  July 21, 1995.}
\thanks{final version: 3/20/1995.}
\thanks{Copyright 1997, AMS.
  Proc. AMS. Vol 125, No 1, Jan 1997, pages 301--308.}


\begin{abstract}
The fundamental lemma is a conjectural identity between the orbital
integrals on two reductive groups.  The fundamental lemma is required
for the stabilization of the trace formula and for various
applications to automorphic forms.  This paper proves the fundamental
lemma for the group $Sp(4)$ and its endoscopic groups.
\end{abstract}

\subjclass[2000]{22E50, 22E35, 20G25}

\maketitle

Let $F$ be a $p$-adic field of characteristic zero with ring of
integers $O_F$.  Let $G$ be an unramified reductive group defined over
$O_F$, and let $H$ be a standard (i.e., untwisted) unramified
endoscopic group of $G$ (also defined over $O_F$).  Fix an embedding
$\xi: {}^LH\to {}^LG$ of $L$-groups, which satisfies the hypotheses
for unramified endoscopic data in [H6,1].  The embedding $\xi$, by the
Satake transform, determines a map $b: \cH_G \to \cH_H$ between the
spherical Hecke algebras $\cH_G$ and $\cH_H$ on $G$ and $H$.  We may
assume that $\cH_G$ and $\cH_H$ are defined relative to the
hyperspecial maximal compact subgroups $G(O_F)$ and $H(O_F)$.

If $f\in C_c^\infty(G)$ and $\gamma_G$ is a strongly regular element
of $G$, we form the orbital integral $\Phi(\gamma_G,f)$ over the
conjugacy class of $\gamma_G$.  Similarly, for $f^H\in C_c^\infty(H)$
and strongly $G$-regular elements $\gamma_H\in H(F)$, we form the
orbital integral $\Phi(\gamma_H,f^H)$.  Various normalizations of
measures enter into the definition of orbital integrals.  Fixing
invariant differential forms $\omega_G$ on $G$ and $\omega_T$ on
$T=C_G(\gamma_G)$ (the centralizer of $\gamma_G$) that are defined
over $F$, we may form the orbital integrals $\Phi(\gamma_G,f)$ with
respect to the measure $|\omega_G/\omega_T|$ on the orbit of
$\gamma_G$.  We assume that the choices $\omega_T$ and $\omega_{T'}$
for various Cartan subgroups $T$, $T'$ are conjugate over the
algebraic closure $\bar F$ of $F$.  We make similar selections of
measures on $H$.

Let $\Delta(\gamma_H,\gamma_G)$ be the Langlands-Shelstad transfer
factor [LS2] with the canonical normalization of [H4,7].
We form the expression
\begin{equation}\label{eqn:1}
\Lambda(\gamma_H,f) = \sum_{\gamma_G}\Delta(\gamma_H,\gamma_G)
\Phi(\gamma_G,f) - \sum_{\gamma'_H} \Phi(\gamma_H',b(f)). % \tag1
\end{equation}
The first sum runs over representatives of all regular semisimple
conjugacy classes in $G$ and the second sum runs over representatives
of the conjugacy classes in $H$ that are stably conjugate to the class
of $\gamma_H$.  Both sums have only finitely many nonzero terms.  Both
$b$ and $\Delta$ depend on the choice of embedding $\xi$.

The measures on $G$ and $H$ must be compatible. This is achieved by
fixing a strongly $G$-regular element $\gamma_H\in H(F)$ such that
both terms of the sum defining $\Lambda(\gamma_H,f)$ are nonzero for
some $f_0\in \cH_G$.  Rescale $|\omega_H|$, so as to modify the second
sum of (1) by a scalar so that $\Lambda(\gamma_H,f_0)=0$.
Equivalently, we may use the normalizations of [H4,14.2].

The following result, known as the fundamental lemma, is conjectured
to hold for any unramified reductive group $G$ in the setting
described above.  It is indispensable to various applications of the
trace formula.

{\bf Theorem.}  Suppose $G = Sp(4)$.  Then $\Lambda(\gamma_H,f)=0$
for all $f\in \cH_G$ and all strongly $G$-regular elements $\gamma_H\in H(F)$.

{\bf Remark.}  The fundamental lemma for $GSp(4)$ follows from this
theorem together with a series of reductions made in [H6,3.6].  The
fundamental lemma for $GSp(4)$ was described in a lecture I gave at
Luminy in 1992 as a consequence of the results of J.-L.  Waldspurger
on the homogeneity of Shalika germs and my earlier work on $GSp(4)$
[H1], [Wa1], [Wa2].  We require the more recent matching results of
[H5] for $Sp(4)$.  A double coset argument discovered by M. Schr\"oder
makes it possible to give another proof of the fundamental lemma for
$GSp(4)$ [S1], [S2].  The details of this approach have been carried
out in a series of preprints by R. Weissauer [We].

A slightly stronger form of the theorem actually holds.  A general
argument of Langlands and Shelstad shows that $\gamma_H$ can be taken
to be any $(G,H)$-regular element [LS3,2.4].

{\bf Proof.}  We will begin with a general reductive group $G$ and
will introduce as hypotheses the parts of the proof of the fundamental
lemma that have been verified only in special cases.  Our approach to
the fundamental lemma is by induction on the dimension of the group.
It is therefore natural to assume that the fundamental lemma is known
for groups of smaller dimension.

Let $K$ be any number field that has a completion at some place $v$
isomorphic to $F$.  Let $G'$ be any reductive group over $K$ with
endoscopic group $H'$ such that at $v$ the pair $(G',H')$ is
equivalent to a proper Levi factor of $G$ together with the endoscopic
group obtained from $H$ by descent, or to the connected centralizer of
a noncentral absolutely semisimple element in $G$ together with the
endoscopic group obtained from $H$ by descent.  Absolutely semisimple
elements, also called ${\Bbb F_q}$-semisimple elements, are defined in
[K].  See also [H4,3].

{\bf Hypothesis 1.}  For any of the reductive groups $G'$ and
endoscopic groups $H'$ obtained as above, the fundamental lemma is
true at almost all places of $K$.

A narrower formulation of this hypothesis is possible.  It is
sufficient to verify the hypothesis for a single carefully chosen
number field $K$ (depending on $F$, $G$, and $H$).  Further, the
fundamental lemma can be reduced to the adjoint group of $G'$,
provided unramified quasicharacters of $G'_{adj}$ are introduced.  See
[H6,3.6].  In the case of $Sp(4)$ the centralizers and Levi factors
all have ranks at most one, so the adjoint group, if not trivial, is
$PGL(2)$.  This special case of the fundamental lemma has several
proofs, for example, [K].

Hypothesis 1 allows us to apply the results of [H6].  In particular,
we may assume that the residual characteristic of $F$ is as large as
we please.  It also follows from [H6] that it is sufficient to prove
that $\Lambda(\gamma_H,1_G)=0$, where $1_G$ is the unit element of the
Hecke algebra $\cH_G$.  Of course, $b(1_G)$ is then the unit element
of $\cH_H$.

By applying descent to Levi factors, we may assume that $H$ is
elliptic [H4,12].

Consider a strongly $G$-regular element $\gamma_H \in G(F)$.  If
$\gamma_H$ is not topologically unipotent, Kazhdan's lemma may be
invoked to reduce to a lower-rank case.  The proof of this in [H4,13]
makes the assumption that $G_{der}$ is simply connected.  This
restriction is unnecessary: the result is proved in [Ko,7.1] without
that restriction.  Thus, we assume that $\gamma_H$ is topologically
unipotent.

Let $\g_{tn}$ be the set of topologically nilpotent elements of the
Lie algebra of $G$.  Let $G_{tu}$ be the set of topologically
unipotent elements of $G$.  Similarly, we consider $\h_{tn} =
\text{Lie}(H)_{tn}$ and $H_{tu}$.  If the residual characteristic is
sufficiently large, then the exponential map gives an analytic
isomorphism between $\g_{tn}$ and $G_{tu}$.

Let $\h'\subset \h$ be the set of strongly $G$-regular elements in
$\h$.  In everything that follows, it is understood that $X_H\in \h'$.
If $X_H\in \h'_{tn} = \h'\cap \h_{tn}$, the transfer factor
$\Delta(\exp(X_H),\gamma_G)$ is zero unless $\gamma_G$ is also
topologically unipotent.  So we may pass to the Lie algebras of $G$
and $H$.

{\bf Hypothesis 2.}  Suppose that the residual characteristic is
sufficiently large.  For $|t|\le 1$, and $X_H\in \h'_{tn}$ fixed,
$\Lambda(\exp(t^2 X_H),1_G)$ is a polynomial in $|t|$.

It is known that for $X_H$ fixed, the transfer factor for $\exp(t^2
X_H)$ is a constant times a power of $|t|$, if the residual
characteristic is not two [H4].  Hypothesis 2 is then true for the
classical groups by the results of Waldspurger, which express orbital
integrals of the unit element on the topologically unipotent set as a
sum of terms, each of which is homogeneous, that is, a monomial in
$|t|$.  See [Wa1], [Wa2,V] for a precise description of the class of
groups treated and the restrictions on the residual characteristic.
As a very special case of Waldspurger's results, we obtain Hypothesis
2 for $Sp(4)$.

Any polynomial in $|t|$ is zero if it is zero for $|t|$ sufficiently
small.  Hypothesis 2 allows us to restrict our attention to a small
neighborhood of the identity where the Shalika germ expansion is
valid.  Consider the Shalika germ expansion of the first sum defining
$\Lambda$ when $X_H$ is sufficiently small in $\h'$.  Set $D(X_H) =
|\prod_\alpha \alpha(X_H)|^{1/2}$, with the sum running over the roots
of $\h$ with respect to a Cartan subalgebra containing $X_H$.  We have
for $f\in C_c^\infty(G)$,
$$D(X_H)\sum_{X_G} \Delta(\exp(X_H),\exp(X_G))\Phi(\exp(X_G),f)=
\sum_{O} \Gamma^G_O(X_H)\mu_O(f),$$ 
for some collection of functions
$\Gamma_{O}^G(X_H)$ defined in a suitable neighborhood of $0$ in
$\h'_{tn}$.  Up to the factor $D$, the left-hand side is the first sum
in (1) defining $\Lambda$.  The functions $\Gamma^G_O$ are
combinations of Shalika germs on $G$.  The sum on the right runs over
unipotent classes in $G$.  We consider this as an identity which
defines $\Gamma_O^G$ in a small neighborhood of $0$ in $\h'$, and then
we extend the functions $\Gamma_O^G$ to all of $\h'$ by homogeneity.

Similarly, we consider a {\it stable} Shalika germ expansion on
$\h'_{tn}$.  For $X_H$ sufficiently small, and $f^H\in C_c^\infty(H)$,
write the second sum in (1) in the form
$$D(X_H)\sum_{X_H'} \Phi(\exp(X_H'),f^H) = \sum_{O'}
\Gamma_{O'}^H (X_H)\mu_{O'} (f^H),$$
where the sum on the right
now runs over unipotent conjugacy classes of $H$.
The functions $\Gamma_{O'}^H$ are stable versions of
Shalika germs.
We extend these functions by homogeneity to $\h'$.

{\bf Hypothesis 3.}  There exists a linear map $b_0:C_c^\infty(G) \to
C_c^\infty(H)$, $f\mapsto b_0(f)$ satisfying
$$\sum_{O} \Gamma^G_O (X_H) \mu_O(f) = \sum_{O'}\Gamma^H_{O'}(X_H)
\mu_{O'}(b_0(f)),$$
for all $X_H\in \h'$.

This hypothesis is called {\it local $\Delta$-transfer at the
  identity\/} in [LS3].  It has been verified for $GSp(4)$ in [H1] and
for $Sp(4)$ in [H5] when the residual characteristic is odd.
Hypothesis 3 is stronger than what is required for the fundamental
lemma.

For each $X_H$, the right-hand side of the identity of Hypothesis 3
defines an invariant distribution on $C_c^\infty(H)$, which is
supported on the unipotent set, namely
\begin{equation}\label{eqn:2}
f^H \mapsto \sum_{O'} \Gamma^H_{O'}(X_H)\mu_{O'}(f^H).
%\tag2
\end{equation}
Let $\cL$ be the span of these distributions.
The linear map $b_0$ gives, by duality, a linear map 
$\mu\mapsto\mu^G$ from $\cL$ 
to invariant distributions
supported on the unipotent set of $G$.  Specifically, for $\mu\in\cL$,
define $\mu^G$ by $\mu^G(f) = \mu(b_0(f))$, for $f\in C_c^\infty(G)$.

{\bf Hypothesis 4.}  $\mu^G(1_G) = \mu(b(1_G))$, for all $\mu \in \cL$.

If $X_H$ is not elliptic, the unipotent distribution $\mu$ on
$C^\infty_c(H)$ defined by (2) satisfies the condition of Hypothesis 4
by descent and induction (Hypothesis 1). See [H4,12].  Call these
distributions {\it stably induced.}

{\bf Proposition.}  If Hypotheses 1, 2, 3, and 4 hold, then the
fundamental lemma is true for $G$ and the endoscopic group $H$.

We emphasize that all four hypotheses are assertions
about what happens almost everywhere for various number
fields $K$.  In Hypothesis 1, this is explicit.  After
Hypothesis 1, we shifted notation, so that $F$ is no
longer a fixed $p$-adic field for which we wish to prove
the fundamental lemma.  It becomes an indefinite $p$-adic field
of sufficiently large residual characteristic, obtained
by completing various number fields $K$.

{\bf Proof.} It is clear by the preceding remarks that
the first two hypotheses reduce the fundamental lemma
to a statement in a small neighborhood of the identity.
We must verify that $b_0(1_G)$ has the same stable
germ expansion as $b(1_G)$, that is, that $\mu(b_0(1_G))=
\mu(b(1_G))$ for all $\mu\in\cL$.  Hypothesis 4 asserts
this, and the result follows. \qed

A unipotent class $O$ is $r$-{\it regular} if $2r=\dim(C_G(u))-
\text{rank}(G)$, for $u\in O$.  The $0$-regular classes are 
regular, and the $1$-regular classes are subregular.
We call the partial sum 
$$\sum_O\Gamma_O^G(X_H)\mu_O(1_G),$$ where the sum ranges
over all $r$-regular unipotent classes $O$ of $G$, the
{\it $r$-regular term\/} on $G$.  Similarly, the corresponding partial
sum
$$\sum_{O'}\Gamma_{O'}^H(X_H)\mu_{O'}(b(1_G))$$
over $r$-regular orbits $O'$ will be called the
{\it stable $r$-regular term\/} on $H$.  This
distinction between {\it term\/} and {\it germ\/} is
crucial. To verify Hypothesis
4, we must show that the $r$-regular terms on $G$ and $H$
coincide for a finite set of elements $\{X_H\}\subset \h'$ whose
distributions (2) span $\cL$.

\bigskip

Now we restrict our attention to the group $G=Sp(4)$.  We may assume
that the residual characteristic is odd.  The elliptic unramified
endoscopic groups of $G$ are $SO(4)$, the quasisplit form $SO^*(4)$ of
$SO(4)$ that splits over an unramified quadratic extension $E/F$, and
the product $SL(2)\times U_E(1)$, where $U_E(1)$ denotes a
one-dimensional nonsplit torus that is split by $E$, again with $E/F$
an unramified quadratic extension.  When $H=SO(4)$, the regular and
both subregular classes give stably induced distributions $\mu$, so
that, by our earlier comment, Hypothesis 4 is satisfied for them.  The
remaining unipotent conjugacy class of $SO(4)$ is treated in Lemma 1.
When $H=SO^*(4)$, the regular class is stably induced. There is no
subregular class with rational points. Again, it is enough to consider
the two-regular class.  If $H=SL(2)\times U_E(1)$, the regular class
is stably induced. It is necessary to consider the subregular
unipotent class.  (In this case, the two-regular germs vanish [H5].)

We take an image of $X_H$ in $Sp(4)$ and let $\pm t_1$ and $\pm t_2$
in $\bar F$ be the eigenvalues of the image in $Sp(4)\subset GL(4)$.

The two-regular term in $GSp(4)$ is a product of three factors:

(i)  The constant $A_1(M)$ of Langlands [L,page 470]

(ii)  The Shalika germ. Formulas appear in [H1,6]

(iii) The unipotent orbital integral $\mu_O(1_G)$ computed
relative to the measure 
$|\omega|=|z\,dz\,dy_1\,dy_2\,dy_3|$ with coordinates
on an open set of $O$:
\begin{equation}\label{eqn:3}
\begin{pmatrix} 1&0&0&0\\ y_1&1&0&0 \\
          y_2&0&1&0 \\ y_3&y_2&-y_1&1 \end{pmatrix}
\begin{pmatrix} 1&0&0&z\\ 0&1&0&0 \\
          0&0&1&0 \\ 0&0&0&1 \end{pmatrix}
\begin{pmatrix} 1&0&0&0\\ -y_1&1&0&0 \\
          -y_2&0&1&0 \\ -y_3&-y_2&y_1&1 \end{pmatrix}
\subseteq O.
%\tag3
\end{equation}
Here $Sp(4)$ is represented as $4\times4$ matrices preserving
the form $x_1\wedge x_4 + x_2\wedge x_3$.

The constant $A_1(M)$ evaluates, by the
formula of Langlands, to $1/2$.  (This factor of $1/2$
stems from the fact that the irreducible divisor that gives
the contribution to the two-regular term has multiplicity 2 in
the Igusa variety.)

We define a constant $[G]$, for any reductive group $G$ over
a finite field $k$, to be the cardinality of $G(k)$ divided by
$q^d$, where $d = \dim(G)$, and $q$ is the cardinality of $k$.
For example, the multiplicative group $\G_m$
gives $[\G_m] = (1-1/q)$, $[SO(4)] = (1-1/q^2)^2$,
and $[Sp(4)] = (1-1/q^2)(1-1/q^4)$. If $G$ is defined over
$F$, we also let $[G]$ denote the corresponding constant
over the residue field $k$ of $F$.
As in [H4,14], we normalize the unit element of the Hecke
algebra of (any) $G$ 
to be $\text{\it ch}/[G]$, where {\it ch\/} is the
characteristic function of $G(O_F)$.  

If $G = SL(2)$, then
the stable subregular term is the product of the Shalika
germ
and the subregular orbital integral. The subregular
orbital integral is merely
$f\mapsto f(1)$.  Let $t=t(X)$ be an eigenvalue of 
$X\in {\mathfrak sl}(2)$.
By [LS1], the subregular term evaluated at $X$ is then
$$|t|I_F/[SL(2)],$$
where the constant $I_F$ is expressed as a principal-value
integral
$$I_F = \int_{Q(F)} |\omega_Q|.$$
The surface $Q$ is a twisted form of a product of two projective
lines.  The volume form $\omega_Q$ is $da\wedge db/(a-b)^2$, for an
appropriate choice of coordinate $a$, $b$.  See [LS1] for details.
The surface $Q$ and integral depend on a choice of Cartan subalgebra
containing $X$, although the notation does not show this.

Since ${\mathfrak so}(4)$ is isomorphic to a direct sum of two copies
of ${\mathfrak sl}(2)$, we may also conclude from the $SL(2)$
calculation that the stable two-regular term of $SO(4)$, evaluated at
$X_H\in {\mathfrak so}(4) '$, is
\begin{equation}\label{eqn:4}
|t_1^2 -t_2^2| I_F^2/[SO(4)].
%\tag4
\end{equation}
Similarly, since ${\mathfrak so}^*(4)$ is a restriction of
scalars of ${\mathfrak sl}(2)$ over $E$, we conclude that the
stable
two-regular term of $SO^*(4)$, evaluated at 
$X_H\in {\mathfrak so}^*(4)'$,
is
\begin{equation}\label{eqn:5}
|t_1^2 -t_2^2| I_E/[SO^*(4)].
%\tag5
\end{equation}
($t_1-t_2$ and $t_1+t_2$ are to be identified with the
positive roots of ${\mathfrak so}(4)$.)
For similar reasons the stable subregular term of
$SL(2)\times U_E(1)$ at $X_H$ is
\begin{equation}\label{eqn:6}
|t_2| I_F/[SL(2)\times U_E(1)].
%\tag6
\end{equation}
($2t_2$ is to be identified with the positive root of ${\mathfrak sl}(2)
\times {\mathfrak u}_E(1)$.)

To prove Hypothesis 4 we must show that these expressions coincide
with the corresponding terms in $Sp(4)$.  We begin with the endoscopic
group $SO(4)$.

{\bf Lemma 1.}  If $H=SO(4)$, then the two-regular term of
$G$, for $X_H$ elliptic, is
$|t_1^2 -t_2^2| I_F^2/[SO(4)]$.

{\bf Proof.}  For this endoscopic group we may rely on the results of
[H1].  In $GSp(4)$ there is a single two-regular unipotent conjugacy
class.

It follows from Hypothesis 3 and Expression 4 that the formula is
correct up to a nonzero scalar. To check the scalar, we pick any
convenient elliptic element $X_H$.  Consider the Cartan subgroup
$(U_E(1)\times U_E(1))/\{\pm1\}$ in $H$, with $E/F$ unramified. Fix
$X_H$ in its Lie algebra.

In the notation of [H1,6], we set
$w' = w_2 \xi_1 \epsilon/f$, where $\epsilon$ is a unit in $E$ of
trace zero.  Also set $\sigma = \sigma_0\sigma_\alpha\sigma_\beta
\sigma_\alpha\sigma_\beta$.  Let
$\eta$ be the unramified quadratic character of $F^\times$.
Equation 6.8 of [H1] gives the
formula
$$|t_1^2 -t_2^2| \int |dw'/w'| |d \ell_2/\ell_2| \eta(1-\ell_2^2)
\int_{Q(F)} |d\xi_1 d\xi_2/(\xi_1-\xi_2)^2|,$$
for the Shalika germ, where $\sigma$ acts on the coordinates
by $\sigma(w')=1/w'$, $\sigma(\ell_2)= \ell_2$, $\sigma(\xi_1)=-1/\xi_2$,
and $\sigma(\xi_2)=-1/\xi_1$.
All integrals extend over ${\Bbb P}^1$ unless indicated
otherwise. The second integral is $I_F$.
Section 11 of [H2] shows that the
first integral is equal to $2I_F$.
The germ is then $2I_F^2|t_1^2 -t_2^2|$.

The integral $\mu_O(1_G)$ is computed by the method of [H2,12].
Details of closely related calculations are also found in [H3,3.9], so
we will simply state the result.  We find
$$\mu_O(1_G) = (1-q^{-2})^{-2} = 1/[SO(4)].$$
The product of the factors (i), (ii), and (iii)
is then
$$|t_1^2 -t_2^2| I_F^2/[SO(4)],$$
as desired.  This completes Lemma 1.
Since this term coincides with the stable two-regular term (4)
of $H$, this also completes the proof of the fundamental
lemma for the endoscopic group $SO(4)$. \qed

Now we turn to the two-regular term 
in the nonsplit case $H = SO^*(4)$. Again it is the
product of three factors: the constant
$A_1(M) = 1/2$, the Shalika germ, and a unipotent
orbital integral $\mu^\kappa_O(1_G)$.
The germ is given by [H5,5.2]. It is
$2 |t_1^2 -t_2^2| I_E$.
Again, by the methods of [H2],
the unipotent orbital integral evaluates to
$$\mu_O^\kappa (1_G) = (1-1/q^2)/[Sp(4)] = 1/[SO^*(4)].$$
(As in [H5], the relevant measure to take is
$$\mu_O^\kappa (1_G) = \int_O \eta (z) |\omega|,$$
where $z$ and $|\omega|$ are as in (3).)
This  is a $\kappa$-orbital
integral on the two-regular conjugacy class in $Sp(4)$. 
The result is two-regular term on $G$
$$|t_1^2 - t_2^2| I_E/[SO^*(4)].$$
This is precisely the stable two-regular
term (5) on $SO^*(4)$.

Finally, we turn to the endoscopic group $H = SL(2) \times U_E(1)$.
We must compare the subregular terms of $G$ and $H$.  The subregular
term on $Sp(4)$ comes from a single $GSp(4)$-orbit (which breaks into
two orbits in $Sp(4)$).  Recall that the subregular unipotent orbits
in $GSp(4)$ are parametrized by quadratic extensions.  The
$GSp(4)$-orbit that enters here is the one that corresponds to the
unramified quadratic extension $E/F$.  We label the two orbits in
$Sp(4)$ as $O_+$ and $O_-$.  We consider the germ for the Cartan
subgroup $U_E(1)\times U_E(1)$ of $Sp(4)$.  In the notation of
[H1,page 230], the Shalika germ is
$$\int|d\xi/\xi| \int \eta(w^2/w_B)|dw/w^2|.$$
Under the substitution $w = x/(t_2 x+t_2)$, this becomes
$$|t_2| \int |d\xi/\xi| 
\int \eta(1-x^2) |dx/x^2| = 2 |t_2| I_F.$$
(The last equality follows from [H2,11].)
The subregular term on $G$ is then 
$$ 2 |t_2| I_F [\mu_{O_+}(1_G) - \mu_{O_-} (1_G)].$$
Rather than evaluate these unipotent orbital integrals directly,
we note that with a different normalization of measures, they
have already been evaluated by Assem [A,2.3.17].  We add primes to
the distributions (and the characteristic function $1_G$) to
indicate Assem's normalizations.  Assem's result is
\begin{equation}\label{eqn:7}
\mu'_{O_+}(1_G') - \mu'_{O_-}(1_G') = {1-q^{-1}\over 2(1+q^{-1})} =
{[\G_m]\over 2[U_E(1)]}.
%\tag7
\end{equation}
(The orbits $O_+$ and $O_-$ are denoted $(\epsilon,1)$
and $(\epsilon,-1)$ in [A].)
To compare the normalizations of measures, we work with a linear
combination that is easier to treat.  The sum over all the subregular
unipotent classes is the Richardson orbit of a parabolic subgroup.
The corresponding distribution $\sum \mu_O$ is easy to compute.
We have
\begin{equation}\label{eqn:8}
{\sum \mu_O (1_G)\over \sum \mu'_O(1'_G)} =
  {\mu_{O_+}(1_G) - \mu_{O_-}(1_G) \over 
   \mu'_{O_+} (1'_G) - \mu'_{O_-} (1'_G)}.
%\tag8
\end{equation}
Assem finds that $\sum\mu_O'(1_G') = 1$.
By parabolic descent we find $\sum\mu_O(f) = \bar f^P(1)$, where
$\bar f^P$ is the usual function on the Levi of $P=MN$ obtained by
descent.  For the function $f = 1_G$, this gives
$\bar f^P(1) = 1/[M] = 1/[SL(2)\times \G_m]$.
Thus, (7) and (8) give
$$\mu_{O_+}(1_G) - \mu_{O_-}(1_G) = 
  {1\over 2} {1 \over [SL(2)\times \G_m]} {[\G_m]\over [U_E(1)]} =
{1\over 2[H]}.$$
The subregular term of $G$ is then
$|t_2|I_F/[H]$.  This is precisely the stable subregular 
term (6)
on $H$.  The proof of the fundamental lemma 
for $Sp(4)$ is complete. \qed

\bigskip
\centerline{\bf References}
\bigskip

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[A]  M. Assem, Unipotent orbital integrals of spherical
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[LS1] R. Langlands, D. Shelstad, On principal values on $p$-adic
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[LS2]  R. Langlands, D. Shelstad,  On the definition of transfer factors, Math. Ann.
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[LS3]  R. Langlands, D. Shelstad,  Descent for transfer factors, The Grothendieck
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[S1]  M. Schr\"oder, Z\"ahlen der Punkte mod $p$ einer
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[S2]  M. Schr\"oder, Calculating $p$-adic orbital integrals on groups
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[Wa1]  J.-L. Waldspurger, Quelques r\'esultats de finitude
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[Wa2] J.-L. Waldspurger, Homog\'en\'eit\'e de certaines
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[We]  R. Weissauer, A special case of the fundamental
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